A prophet inequality for $$L^p$$ L p -bounded dependent random variables

A prophet inequality for $$L^p$$ L p -bounded dependent random variables Let $$X=(X_n)_{n\ge 1}$$ X = ( X n ) n ≥ 1 be a sequence of arbitrarily dependent nonnegative random variables satisfying the boundedness condition $$\begin{aligned} \sup _\tau \mathbb {E}X_\tau ^p\le t, \end{aligned}$$ sup τ E X τ p ≤ t , where $$t>0$$ t > 0 , $$1<p<\infty $$ 1 < p < ∞ are fixed numbers and the supremum is taken over all finite stopping times of $$X$$ X . Let $$M=\mathbb {E}\sup _n X_n$$ M = E sup n X n and $$V=\sup _\tau \mathbb {E}X_\tau $$ V = sup τ E X τ denote the expected supremum and the optimal expected return of the sequence $$X$$ X , respectively. We establish the prophet inequality $$\begin{aligned} M\le V+\frac{V}{p-1}\log \left( \frac{te}{V^p}\right) \end{aligned}$$ M ≤ V + V p - 1 log t e V p and show that the bound on the right is the best possible. The proof of the inequality rests on Burkholder’s method and exploits properties of certain special functions. The proof of the sharpness is somewhat indirect, but we also provide an indication how the extremal sequences can be constructed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

A prophet inequality for $$L^p$$ L p -bounded dependent random variables

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Publisher
Springer Basel
Copyright
Copyright © 2014 by The Author(s)
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0295-3
Publisher site
See Article on Publisher Site

Abstract

Let $$X=(X_n)_{n\ge 1}$$ X = ( X n ) n ≥ 1 be a sequence of arbitrarily dependent nonnegative random variables satisfying the boundedness condition $$\begin{aligned} \sup _\tau \mathbb {E}X_\tau ^p\le t, \end{aligned}$$ sup τ E X τ p ≤ t , where $$t>0$$ t > 0 , $$1<p<\infty $$ 1 < p < ∞ are fixed numbers and the supremum is taken over all finite stopping times of $$X$$ X . Let $$M=\mathbb {E}\sup _n X_n$$ M = E sup n X n and $$V=\sup _\tau \mathbb {E}X_\tau $$ V = sup τ E X τ denote the expected supremum and the optimal expected return of the sequence $$X$$ X , respectively. We establish the prophet inequality $$\begin{aligned} M\le V+\frac{V}{p-1}\log \left( \frac{te}{V^p}\right) \end{aligned}$$ M ≤ V + V p - 1 log t e V p and show that the bound on the right is the best possible. The proof of the inequality rests on Burkholder’s method and exploits properties of certain special functions. The proof of the sharpness is somewhat indirect, but we also provide an indication how the extremal sequences can be constructed.

Journal

PositivitySpringer Journals

Published: Jun 11, 2014

References

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